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Theorem canthwdom 9576
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 9132, equivalent to canth 7358). (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
canthwdom ¬ 𝒫 𝐴* 𝐴

Proof of Theorem canthwdom
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0elpw 5347 . . . . 5 ∅ ∈ 𝒫 𝐴
2 ne0i 4329 . . . . 5 (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅)
31, 2mp1i 13 . . . 4 (𝒫 𝐴* 𝐴 → 𝒫 𝐴 ≠ ∅)
4 brwdomn0 9566 . . . 4 (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴))
53, 4syl 17 . . 3 (𝒫 𝐴* 𝐴 → (𝒫 𝐴* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴))
65ibi 267 . 2 (𝒫 𝐴* 𝐴 → ∃𝑓 𝑓:𝐴onto→𝒫 𝐴)
7 relwdom 9563 . . . . 5 Rel ≼*
87brrelex2i 5726 . . . 4 (𝒫 𝐴* 𝐴𝐴 ∈ V)
9 foeq2 6796 . . . . . . 7 (𝑥 = 𝐴 → (𝑓:𝑥onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝑥))
10 pweq 4611 . . . . . . . 8 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
11 foeq3 6797 . . . . . . . 8 (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
1210, 11syl 17 . . . . . . 7 (𝑥 = 𝐴 → (𝑓:𝐴onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
139, 12bitrd 279 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝑥onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
1413notbid 318 . . . . 5 (𝑥 = 𝐴 → (¬ 𝑓:𝑥onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴onto→𝒫 𝐴))
15 vex 3472 . . . . . 6 𝑥 ∈ V
1615canth 7358 . . . . 5 ¬ 𝑓:𝑥onto→𝒫 𝑥
1714, 16vtoclg 3537 . . . 4 (𝐴 ∈ V → ¬ 𝑓:𝐴onto→𝒫 𝐴)
188, 17syl 17 . . 3 (𝒫 𝐴* 𝐴 → ¬ 𝑓:𝐴onto→𝒫 𝐴)
1918nexdv 1931 . 2 (𝒫 𝐴* 𝐴 → ¬ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴)
206, 19pm2.65i 193 1 ¬ 𝒫 𝐴* 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1533  wex 1773  wcel 2098  wne 2934  Vcvv 3468  c0 4317  𝒫 cpw 4597   class class class wbr 5141  ontowfo 6535  * cwdom 9561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-wdom 9562
This theorem is referenced by:  pwdjudom  10213
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